Jiang's second-smallest alpha invariant conjecture for K-semistable Fano manifolds

Let $X$ be a K-semistable Fano manifold of dimension $n$, and let $\alpha(X)$ denote its alpha invariant.

Jiang's conjecture. $\alpha(X)<\frac{1}{n}$ if and only if $X\cong \mathbb{P}^n$.

This conjecture asks whether projective space is the unique K-semistable Fano manifold whose alpha invariant lies below $\frac{1}{n}$. It concerns the gap between the sharp general lower bound $\alpha(X)\geq \frac{1}{n+1}$ and the next possible alpha invariant; the paper's construction of examples with alpha invariant $\frac{2}{2n+1}$ addresses the related question of whether values strictly between $\frac{1}{n+1}$ and $\frac{1}{n}$ occur.